Observation of the Goos-Hänchen shift in graphene via weak measurements
Shizhen Chen, Chengquan Mi, Liang Cai, Mengxia Liu, Hailu Luo,∗ and Shuangchun Wen
Laboratory for spin photonics, School of Physics and Microelectronics Science, Hunan University, Changsha 410082,China
(Dated: September 5, 2016)
We report the observation of the Goos-Hänchen effect in graphene via a weak value amplification
scheme. We demonstrate that the amplified Goos-Hänchen shift in weak measurements is sensitive
to the variation of graphene layers. Combining the Goos-Hänchen effect with weak measurements
may provide important applications in characterizing the parameters of graphene.
arXiv:1609.00456v1 [physics.optics] 2 Sep 2016
Keywords: Polarization, Optics at surfaces, Instrumentation, metrology
The behavior of plane wave in reflection can be simply predicted by geometrical optics. However, for the
bounded beam of light, it may undergo extra shifts due
to the occurrence of diffractive corrections. Such shifts
are known as the Goos-Hänchen (GH) [1] and the ImbertFedorov [2, 3] shifts according to the directions parallel
and perpendicular to the plane of incidence, respectively.
In recent years, the research of the GH shift is still active
although it was discovered more than 60 years ago. The
corresponding studies reach not only to the investigation
of the inherent physics behind this phenomenon [4–8],
but also to the behavior of the shift at various reflecting
surfaces [9–14], especially the theoretical works of the GH
shift in reflection from a graphene-dielectric interface.
The system with graphene is very interesting for the observation of beam shifts due to its flexible optical properties. The Fresnel reflection coefficients with the existence
of graphene become different [15, 16], and the behavior of
the GH shift is changed or tunable [17, 18]. In particular,
the GH shift on a substrate coated with graphene can be
quantized in an external magnetic field [19]. Even though
in a common environment, a giant GH shift in graphene
was observed recently [20].
In this paper, we investigate the GH shift in graphene
under a total internal reflection (TIR) condition. Like
other beam shifts, the GH is generally very small and
difficult to be directly observed. Here we amplify it via a
weak measurement technique to overcome this difficulty.
Interestingly, we find that the shift amplified by a weak
value scheme is sensitive to the graphene layers. Weak
measurements are an important and convenient approach
that has reached fruitful achievements for detecting light
beam shifts [21–24]. This novel conception was first proposed in the context of quantum mechanics and then has
been extensively studied [25–28]. With the help of weak
measurements, the GH shift in TIR [29] or in partial reflection [30] can be observed recently. In our work, the
GH shift occurs in the regime of TIR (the reflected intensity is equal to incident intensity), and the success rate
of the postselection is still large due to no requirement
of a very large weak value. That is, the final output intensity is still strong and measured data are stable. Our
result suggests that this technique may become an alter-
∗ Electronic
address: hailuluo@hnu.edu.cn
FIG. 1: (color online) Schematic illustrating the GH shift in
graphene under TIR. A 45◦ linearly polarized beam labeled
by |Ai hits a glass-graphene-air interface at an incident angle
θi . Then the components |Hi and |V i experience lateral small
GH shifts DH and DV , respectively.
native way to effectively and conveniently identify layers
of few-layer graphene.
Consider a reflection system shown in Fig. 1. An incoming beam is at the 45◦ linearly polarization state |Ai.
Under a TIR condition, the reflected light consists of two
beams, one is the horizontal (H) component |Hi displaced by DH and the other is the vertical (V ) component |V i displaced by DV . We only concern about the
difference between DV and DH because it is the measured variable in our weak measurements scheme. To
calculate the GH shift (DV − DH ) for different graphene
layers, the obtainment of the reflection coefficient in the
graphene-dielectric interface is important. The reflection
coefficient here is given by [20, 34]
′
RA + RA exp (2idkgz )
rA =
.
′
1 + RA RA exp (2idkgz )
′
(1)
Here, RA and RA are the Fresnel reflection coefficients
in the glass-graphene and graphene-air interfaces, respectively, A ∈ {H, V }. kgz is the component of wave vector
k0 in graphene alone z direction. d = ζ∆d is the thickness
of the graphene film with ζ and ∆d = 0.34nm representing the layer numbers and the thickness of single layer
graphene, respectively. In TIR, the reflection coefficient
rA is complex and can be written as rA = |rA | exp(iϕA ).
Neglecting the small angular shift due to the slow varia-
2
cate here due to the effect of reflection coefficients. In
the experiment, we set the optical axis of GLP1 to
45◦√to project
the incident polarization state on |ii =
√
(1/ 2, 1/ 2). And this polarization state changes to
|γζ i = F |ii when the light is reflected at the interface. F
is the reflection matrix [32]
−rH 0
F =
,
(3)
0 rV
of which action is also a part of the preselection process, and the preselected state for our weak measurements scheme is |γζ i. Note that the state |γζ i is an elliptical polarization state which is different for one, two,
and three graphene layers. Then we consider the weak
coupling. The tiny GH effect is regarded as a weak measuring process here, as labeled by the dashed circle in
Fig. 2(c). In the language of quantum mechanics, this
effect described by operator is [6, 30, 31]
ˆ = DH 0 .
GH
(4)
0 DV
FIG. 2: (Color online) (a) (DV -DH ) as a function of incident
angle for different graphene layers. The red dashed line indicates the incident angle (42◦ ) in our experiment, the angle
of total reflection is 41.3◦ . (b) Representation on the Bloch
sphere of the states |ii, |γζ i, and |f i. |ii is the incident state
and |f i is the postselected state. |γζ i is the preselected state
after reflecting at the graphene-dielectric interface. The state
|γζ i is off to be antiparallel to |ii in two angular directions depending on the graphene layer numbers. The red dashed circle
represents the trajectory of |f i when we rotate the half-wave
plate (HWP). (c) Experimental setup: a Gaussian beam generated by a He-Ne laser (632.8nm, Thorlabs HNL210L-EC).
GLP1 and GLP2, Glan Laser polarizers; QWP, quarter-wave
plate; L1 and L2, lenses with focal length 125mm and 250mm
respectively. The beam waist after L1 is 71.25 µm. The data
are detected by a CCD camera (Coherent LaserCam HR).
Insets show the rotations of GLP1, QWP, and GLP2.
tion of |rA | [15], the GH shifts for H and V components
are spatial and can simply form as
DA = −
1 ∂ϕA
.
nk0 ∂θi
(2)
Here, n = 1.515 is the refractive index of prism. From
Eq. (2), we plot the curves of the GH shift as a function
of incident angle for different graphene layers with the
refractive indexes (3 + 1.149i) of graphene, as shown in
Fig. 2(a). The shift decreases with increasing numbers of
layer, but the decrement is tiny and all shifts are small.
To amplify these small shifts, the weak measurements
are employed and the experimental setup is plotted in
Fig. 2(c). This setup is similar to that prescribed in [29].
In a schema of weak measurements, the preselected state,
postselected state, and weak coupling between the system
and pointer are three key elements. The corresponding
elements in our scheme will be clear in the following.
We first discuss the preselected state which is deli-
We next analyze the postselection, which can be realized
by the combination of QWP, HWP, and GLP2. In the
experiment, the optical axis of QWP is fixed to 45◦ from
the x axis, and the rotation angle of HWP is α. These
settings described by the Jones matrices are
1
1 −i
QWP = √
2 −i 1
cos(2α) sin(2α)
.
(5)
HWP =
sin(2α) − cos(2α)
The optical axis of GLP2 is (45◦ ± ∆) to project on the
state
hGLP2| = cos(45◦ ± ∆) sin(45◦ ± ∆) .
(6)
Putting Eqs. (5) and (6) together, we obtain the postselected state as
hf | = ei(2α∓∆) −e−i(2α∓∆) .
(7)
For convenience, we represent above states |ii, |γζ i, and
|f i on the Bloch sphere, as shown in Fig. 2(b). One can
see that the postselected state |f i is limited on the red
dashed circle by adjusting HWP [from Eq. (7)]. For the
preselected state |γζ i, it exhibits a deviation from the
red dashed circle on the Bloch sphere in Fig. 2(b) [see
Fig. 3(a) for the cases of different layers]. The reason
for this deviation is that |rH | 6= |rV | and the difference
between |rH | and |rV | increases with the increased layers
of graphene.
With the preselected and postselected states discussed
above, the weak value in our weak measurements can be
obtained as
ˆ ζi
hf |GH|γ
1
Aw
= (DH + DV ) +
(DH − DV ),
hf |γζ i
2
2
(8)
3
Amplified shift (µm)
(a) 400
200
0
No graphene
One-layer
Two-layer
Three-layer
-200
-400
-4
(b)
where Aw = hf |σˆ3 |γζ i/hf |γζ i, and σˆ3 is the Pauli matrix.
Obviously, the weak value from Eq. (8) is related to ζ,
i.e., the layer of graphene. Here, Aw in all cases is a
complex number except the one of ζ = 0, in which it
is pure imaginary, for the full study in different theory
about this case one can see [29]. Note that the GH shift
in TIR is a spatial shift (the angular GH shift is too
small even in graphene), and the imaginary part of Aw
can naturally convert the relevant spatial shift (DH −
DV ) into an angular one. Therefore, in order to obtain
the centroid position hxi of the beams, the propagation
effect in all cases should be considered [4, 21]. Containing
Eq. (8), hxi is obtained as
1
z
[(DH +DV )+Re(Aw )(DH −DV )+ Im(Aw )(DH −DV )],
2
zr
(9)
where zr is the Rayleigh range and z is the propagation.
In fact, the shift we measure in the experiment is a relative position of the beams on CCD for the postselected
states with ±∆. Thus, the third term in Eq. (9) is pivotal
and the first term is irrelative. The second term mainly
results from the inequality between |rH | and |rV |.
We now turn to the procedure for experimentally observing amplified GH shift. The rotations of GLP1 and
QWP are all fixed at 45◦ . We first set GLP2 to 45◦ , i.e.,
∆ = 0. Then we rotate the HWP with an angle α and the
state |f i will project on the red dashed circle in Fig. 2(b).
Note that the rotation α is different for different layers of
graphene, but it is unnecessary to clarify it in the experiment. We adjust the HWP until the output intensity on
CCD becomes minimize, which indicates that the postselected state |f i is closest orthogonal to the preselected
0
∆ (degrees)
2
4
G
One-layer
Two-layer
Three-layer
Intensity (a.u.)
FIG. 3: (Color online) (a) Representation on the Bloch sphere
of the state |γζ i and |f i when the tunable state |f i is closest orthogonal to |γζ i (reading intensity on CCD becomes
minimum). The yellow area indicates the deviation to the
circle of |f i. (b) The theoretical minimum intensity by adjusting HWP. (c) The corresponding minimum intensity we
read out from CCD. Note that the first, second, third, and
fourth columns correspond to the cases of no, one-layer, twolayer, and three-layer graphene, respectively.
-2
2D
1200
1500
1800
2100
2400
-1
Raman shift (cm )
2700
FIG. 4: (Color online) (a) The amplified shift as the function
of angle ∆ for different graphene layers. Experimental data
are shown as open dots with error bars. (b) Raman spectrum of our samples for one-layer, two-layer, and three-layer
graphene.
state |γζ i. After minimizing the output intensity, we rotate the GLP2 first to (45◦ + ∆) and then to (45◦ − ∆)
to measure the final amplified shift.
As discussed above, we use a quantum mechanical description to analyze the amplified GH shift in order to
provide a good physical insight and simplify the analysis. In fact, the process for the weak measurement of
GH effect can be described by using standard wave optics [29], and the simulative minimum output intensity is
illustrated in Fig. 3(b). We see that only in the case of
no graphene the minimum intensity exhibits double-peak
profile, which is a common distribution of minimum output intensity in weak measurements [26]. This is because
in this case the state |f i is possible to be orthogonal to
|γ0 i. For other cases with the existence of graphene, the
tunable postselected state can not be orthogonal to the
preselected state, and the nonorthogonal degree becomes
larger when the layers of graphene increase, leading to
a smaller weak value. As a result, the minimum intensity tends to a Gaussian form [33]. The corresponding
minimum intensity we experimentally observe is shown
as Fig. 3(c).
We measure the amplified GH shifts in no, one-layer,
4
two-layer, and three-layer graphene. The theoretical
results of the amplified shift are briefly described by
Eq. (9). However, in order to avoid the approximate
limits and get accurate values, the curves in Fig. 4(a)
are given by a precise weak measurement theory [27].
For a fixed angle ∆, the amplified shift decreases with
increasing layers. Thus, we can conveniently determine
the layers of a sample at a special angle. In our experiment, each sample we fabricate is uniform layer and the
size of sample is 1cm∗1cm. In practice, we repeat the experiment of weak measurements several times in different
measuring place for each sample and the data are nearly
same. To confirm the corresponding layers of graphene
film, we provide their Raman spectra in Fig. 4(b). The
layers of each sample deduced from our data coincide well
with the results from Raman spectra.
The measurability of very small displacements is ultimately limited by the quantum noise of the light, because
enough photons need to be collected to resolve the position of the field [21]. Note that another interesting beam
shift induced by photonic spin Hall effect can also be used
to identify graphene layers [34]. In that case, the experimental measurement was performed near Brewster angle. Therefore, the experimental data are a little unstable
due to a low reflection intensity near Brewster angle. In
present case, however, enough photons can be captured
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Acknowledgments
This research was supported by the National Natural
Science Foundation of China (Grants Nos. 11274106 and
11474089); Hunan Provincial Innovation Foundation for
Postgraduate (CX2016B099).
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